A counterexample to the Hirsch conjecture

@article{Santos2010ACT,
  title={A counterexample to the Hirsch conjecture},
  author={F. Santos},
  journal={ArXiv},
  year={2010},
  volume={abs/1006.2814}
}
  • F. Santos
  • Published 2010
  • Mathematics, Computer Science
  • ArXiv
The Hirsch Conjecture (1957) stated that the graph of a d-dimensional polytope with n facets cannot have (combinatorial) diameter greater than n d. That is, any two vertices of the polytope can be connected by a path of at most n d edges. This paper presents the rst counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope with 48 facets that violates a certain generalization of the d-step conjecture of Klee and Walkup. 
Recent progress on the combinatorial diameter of polyhedra and simplicial complexes
  • F. Santos
  • Computer Science, Mathematics
  • SoCG '13
  • 2013
The diameters of network-flow polytopes satisfy the Hirsch conjecture
Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets
On the Circuit Diameter Conjecture
Polyhedral graph abstractions and an approach to the Linear Hirsch Conjecture
  • Edward Kim
  • Mathematics, Computer Science
  • Math. Program.
  • 2014
A simple proof of tail--polynomial bounds on the diameter of polyhedra
Pushing the boundaries of polytopal realizability
Circuit diameter and Klee-Walkup constructions
An Improved Kalai-Kleitman Bound for the Diameter of a Polyhedron
  • M. Todd
  • Mathematics, Computer Science
  • SIAM J. Discret. Math.
  • 2014
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 94 REFERENCES
An Update on the Hirsch Conjecture
Many Polytopes Meeting the Conjectured Hirsch Bound
  • F. Holt, V. Klee
  • Mathematics, Computer Science
  • Discret. Comput. Geom.
  • 1998
Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets
The Monotonic Bounded Hirsch Conjecture is False for Dimension at Least 4
  • M. Todd
  • Mathematics, Computer Science
  • Math. Oper. Res.
  • 1980
The d-Step Conjecture and Its Relatives
Decompositions of Simplicial Complexes Related to Diameters of Convex Polyhedra
Thed-step conjecture for polyhedra of dimensiond<6
On the diameter of convex polytopes
Embedding a Pair of Graphs in a Surface, and the Width of 4-dimensional Prismatoids
A quasi-polynomial bound for the diameter of graphs of polyhedra
...
1
2
3
4
5
...